classical integrable structures in quantum integrable models joint work with v.kazakov and a.sorin...
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CLASSICAL INTEGRABLECLASSICAL INTEGRABLESTRUCTURES IN QUANTUM STRUCTURES IN QUANTUM
INTEGRABLE MODELSINTEGRABLE MODELS
joint work with V.Kazakov and A.Sorin
Leiden, 14 April 2010
based on
A.Zabrodin (ITEP, Moscow)
(and earlier works with I.Krichever, O.Lipan and P.Wiegmann)
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Classical soliton equations in quantum integrable models at :
Correlation functions(Wu, McCoy, Tracy, Barouch; Perk;Sato, Jimbo, Miwa; Izergin, Korepin, Its, Slavnov;…)
Functional relations for quantum transfer matrices(Bazhanov, Kulish, Reshetikhin; Klumper, Pearce;Kuniba, Nakanishi, Suzuki; Tsuboi;…)
Partition functions of 2D lattice modelswith domain wall boundary conditions(Izergin; Slavnov; Foda, Wheeler, Zuparic;…)
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Theory of solitons
Non-linear integrable PDE’s (KdV, KP, sine-Gordon, Toda,…)
Hierarchies
Integrable discretization
Hirota equation
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Quantum integrability (quantum inverse scattering method)
R-matrix, RRR=RRR
T-matrix, TT R =RTT
T-matrix, T = tr T , [T,T]=0
Fusion relationTT-TT+TT=0
Bethe equations,spectrum
Q-operator
Algebraic Bethe ansatz
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Soliton equations
Hirota equation
R-matrix
T -matrix
T-matrix
Fusion rules
Q-operator
Alg. Bethe ansatz
Bethe eq-s
Auxiliary linearproblems
Polynomialsolutions
Classical world
Quantum world
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Plan1.1. Integrable SUSY Integrable SUSY GL(K|M)-GL(K|M)-invariant spin chains invariant spin chains
2. From 2. From RR-matrix to Hirora equation (-matrix to Hirora equation (TTTT-relation)-relation)
- Boundary and analytic conditions- Boundary and analytic conditions
5. 5. QQ-QQ-relationrelation ( (Hirota equation forHirota equation for Q’s) Q’s) and Bethe equationsand Bethe equations
3. Solving the 3. Solving the TTTT-relation by classical methods-relation by classical methods
- “Undressing” procedure: nested Bethe ansatz - “Undressing” procedure: nested Bethe ansatz as a chain ofas a chain of Backlund transformationsBacklund transformations
- Auxiliary linear problems- Auxiliary linear problems
- Backlund transformations- Backlund transformations
4. Generalized Baxter’s 4. Generalized Baxter’s TQ-TQ-relationsrelations
- Functional relations for quantum transfer matrices- Functional relations for quantum transfer matrices
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R-matrix
Physical interpretations:
- S-matrix of an elementary scattering process
- Matrix of Boltzmann weights at a vertex
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Yang-Baxter (RRR=RRR) relation
1
1
2
2
3 3
=
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Example: rational R-matrices
is the permutation matrix
More general:
is a representation of a group
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Quantum monodromy matrix
auxiliary space
quantum space
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(P.Kulish 1985)(P.Kulish 1985)
obeys the graded Yang-Baxter equation
--invariantinvariant -matrix acts in the tensorproduct of two irreps and
obeys the conditionfor any
(P.Kulish, E.Sklyanin 1980)
GL(K|M) spin chains
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spectral parameter
parity
generators of
arbitrary, vector representation
Cin the space C
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Quantum monodromy matrix
auxiliary space
quantum space
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Quantum transfer matrix
Yang-Baxter equation commutativity
(periodic b.c.)
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Rectangular representations (We consider covariantrepresentations only)
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Functional relationsfor transfer matrices
P.Kulish, N.Reshetikhin 1983; V.Bazhanov, N.Reshetikhin 1990;A.Klumper, P.Pearce 1992;A.Kuniba, T.Nakanishi, J.Suzuki 1994;Z.Tsuboi 1997 (for SUSY case)
for general irreps
through for the rows
can be expressed
sIn particular,
This is the Bazhanov-Reshetikhin det-formula for rectangular irreps.
Analog of Weyl formula for characters
The same for SUSY and ordinary case
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TT-relation
T (u+1)T(u-1) T(u) T (u)
T(u)
T (u)
s
a
directly follows from the BR det-formula
is identical to the HIROTA EQUATION
All T-operators can be simultaneously diagonalized
The same relation holds for any of their eigenvalues
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Hirota difference equation (R.Hirota 1981)
Integrable difference equation solvable by classicalinverse scattering method
A ‘’master equation” of the soliton theory:
provides universal discretization of integrable PDE’s
generates infinite hierarchies of integrable PDE’s
We use it to find all possible Baxter’s TQ-relations and nested Bethe ansatz equations for super spin chains
(similar method for ordinary spin chains: I.Krichever, O.Lipan, P.Wiegmann, A.Zabrodin 1996)
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Analytic and boundary conditions
are polynomials in
length of spin chain
for any a,sof degree Nu
What is And ??
Transfer matrix for the trivial irrep is proportional to the identity operator:
Here a,s > 0.Extension tonegative values?
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Fat hook The domain where T(a,s,u) for GL(K|M)spin chains does not vanish identically
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SUSY boundary conditions (Z.Tsuboi 1997)
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The standard classical scheme
Nonlinear integrableequation
Compatibility of auxiliary linear problems
Solutions to thelinear problems
Solutions to the nonlinear eq-n
We are going to apply this scheme to the Hirota equation for transfer matrices
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Auxiliary linear problems (Lax pair)
The Hirota equation for T(a,s,u)
for any z (classical spectral parameter)
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F(u)
T(u+1)
T(u+1)
F(u)
T(u+1)
F(u)
T(u) F(u+1)
T(u)
F(u+1)
T(u+1)F(u)
s
a
_
_ z
=
=
z
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Backlund transformations
Simple but important fact:
FF obeys the same Hirota equation obeys the same Hirota equation
T Fis a is a Backlund transformationBacklund transformation
Therefore,
Boundary conditions for F - ?
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T
F
K
M M+1
K
K-1
M
There are two possibilities:
1) K K-1
2) M M+1
We need inverse transform!
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Undressing by BT-I: vertical move
K
K-1
M
T(a,s,u) ≡ TK,M (a,s,u) → F(a,s,u) ≡ TK-1,M(a,s,u)
gl(K|M) gl(K-1|M)
∩
Notation:
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Undressing by BT-II: horizontal move
K
M-1 M
gl(K|M) gl(K|M-1)
∩
T(a,s,u) ≡ TK,M (a,s,u) → F(a,s,u) ≡ TK,M-1(a,s,u)Notation:
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Repeating BT-I and BT-II several times, we introduce the hierarchy of functions
k = 1,… , K;m = 1,… , M
such that
They obey the Hirota equation at anyThey obey the Hirota equation at any “ “levellevel” k,m” k,mand are polynomials inand are polynomials in u u for anyfor any a, s, k, m a, s, k, m
They are connected by the Backlund transfromationsThey are connected by the Backlund transfromations
trivial solution in thedegenerate domain
At the highest levelAt the highest level
At the lowest levelAt the lowest level
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The idea is to “undress” the problem to the trivial oneusing a chain of Backlund transformations
At each step we have a continuous parameter z
Put for BT-I
for BT-II
“Undressing” to :
In the following we set
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BT-I:
BT-II:
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Boundary conditions at intermediate levelsk=1, … , K, m=1, … , M
a
s
k
m
fixed polynomial
eigenvalues of Baxter’s Q-operators
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Operator generating series
Pseudo-difference operator
Similar object at any level k,m:
for normalization
It is clear that
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Operator form of the Backlund transformations
Backlund transformations BT-I and BT-II can be represented asrecurrence relations for the
where
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Factorization formulas
Ordered product:
(K,M)
(0,0) (0,M)
k
m
(K,0) (K,M)
(0,0)
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Arbitrary undressing path
(K,M)
(0,0) m
kEach step brings
Each step brings
product of these factors along the path according to the order of the steps
Equivalence of these representations follows from a discrete zero curvature condition
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“Duality” transformations Z.Tsuboi 1997;N.Beisert, V.Kazakov, K.Sakai, K.Zarembo 2005
fermionic root
The “duality” transformations can be most clearly understoodThe “duality” transformations can be most clearly understoodin terms of in terms of zero curvature conditionzero curvature condition on the on the (k,m) (k,m) lattice andlattice and thethe QQ-relationQQ-relation (V.Kazakov, A.Sorin, A.Zabrodin 2007)(V.Kazakov, A.Sorin, A.Zabrodin 2007)
Undressing path SUSY Dynkin diagram
elementarytransformation
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“Zero curvature”
The QQ-relation
Again the Hirota equation!Again the Hirota equation!
We need polynomial solutions
with the “boundary conditions”
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Building blocks for Bethe equations
k-1,m
k,m
k+1,m
k,mk,m-1 k,m+1
k,m
k-1,m
k,m+1
k,m
k+1,m
k,m-1
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Bethe equations for arbitrary path
(K,M)
(0,0) m
k
2
1
3 4
5
…6
1 for
-1 for n
n
a=1, …, K+M-1
is the Cartan matrix
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Conclusion
A lesson: from the point of view of the discreteHirota dynamics the SUSY case looks more natural and transparent
Our approach provides an alternative to algebraic Bethe ansatz